Solving Demand Curves: Finding P And Q

Jan 12, 2026 by Tim Redaksi 39 views

Hey guys! Let's dive into a classic economics problem. We're gonna figure out how to find the values of P and Q (price and quantity, respectively) using a demand curve equation. The cool thing is, we're not just solving the equation; we're going to use a little trick to make things easier. Ready to get started? Let's break it down step by step.

Understanding the Demand Curve

So, what exactly is a demand curve? Simply put, it's a line on a graph that shows how the quantity of a good or service that people want to buy (Q) changes as the price (P) changes. Generally, as the price goes up, the quantity demanded goes down, and vice-versa. The equation we're given, QP - 10P - 10Q + 80 = 0, represents a specific demand curve. Our mission, should we choose to accept it, is to find the values of P and Q that satisfy this equation. In this case, we're given some possible answers, so we're going to see which one works. This is like a puzzle, and we have to find the piece that fits just right.

Now, the equation might look a little intimidating at first glance, but don't worry! We'll use a method that makes it much more manageable. The key is to rearrange the equation to isolate either P or Q. This way, we can see the relationship between the two variables more clearly. The process will involve some algebraic manipulation – nothing too complicated, I promise! We're essentially going to transform the equation into a form that's easier to work with, allowing us to find the correct values for P and Q. Think of it as preparing a recipe: you need to gather all the ingredients and follow the steps to create a delicious outcome. In this case, our 'delicious outcome' is finding the correct price and quantity.

The Equation and Our Goal

Alright, let's take a look at our equation again: QP - 10P - 10Q + 80 = 0. Our task is to determine which of the provided options – A, B, C, or D – correctly satisfies this equation. Each option gives us a pair of P and Q values. To find the right answer, we'll substitute the P and Q values from each option into the equation and see which one makes the equation true. If both sides of the equation are equal after the substitution, then we've found our answer! It's kind of like a detective story: we're given clues (the possible answers) and we're looking for the one that fits the evidence (the equation). The goal here is to carefully substitute and check, making sure we don't miss any steps. This method is straightforward, and with a little patience, we'll get there.

We also have a 'hint' to help us – the phrase 'right and left sides plus 20'. This suggests that we should add 20 to both sides of the equation. Why? Because sometimes, this simple move can help to simplify the equation, making it easier to solve. Always remember that whatever you do to one side of an equation, you must do to the other side to keep it balanced. This fundamental principle of algebra is key to successfully solving equations.

Solving for P and Q

Let's go through the possible answers one by one. Remember, we're substituting the values of P and Q from each option into the equation QP - 10P - 10Q + 80 = 0 to see which one works. We'll start with option A: P = 8, Q = 10. Substituting these values, we get: (8 * 10) - 10 * 8 - 10 * 10 + 80 = 80 - 80 - 100 + 80 = -20. This is not equal to zero, so option A is incorrect. Next up, option B: P = 10, Q = 10. Substituting these values into the original equation: (10 * 10) - 10 * 10 - 10 * 10 + 80 = 100 - 100 - 100 + 80 = -20. Again, this does not equal zero, therefore option B is also not the correct answer.

Now, for option C: P = 10, Q = 8. Plugging these values: (10 * 8) - 10 * 10 - 10 * 8 + 80 = 80 - 100 - 80 + 80 = -20. Once again, this does not equal zero. So, unfortunately, option C is also incorrect. And finally, let's test option D: P = 10, Q = 6. Substituting into the original equation: (10 * 6) - 10 * 10 - 10 * 6 + 80 = 60 - 100 - 60 + 80 = -20. This does not equal zero either. But hold on, the right answer is not in the options.

There must be an error somewhere in this problem. It seems none of the answers provided is correct. Let's try to factor the original equation. First of all, the hint says we must 'plus 20' to the right and left side. Since the right side is 0, by adding 20, we will have:

QP - 10P - 10Q + 80 + 20 = 20

QP - 10P - 10Q + 100 = 20

However, it's very difficult to factor this one with the given options. Let's analyze the options given, and if they will fit the original equation or not.

For option A: P = 8, Q = 10. Substituting these values, we get: (8 * 10) - 10 * 8 - 10 * 10 + 80 = 80 - 80 - 100 + 80 = 0. We can say it's approximately 0, so let's check it in the modified equation. Since it will equal -20, option A is wrong.

For option B: P = 10, Q = 10. Substituting these values into the original equation: (10 * 10) - 10 * 10 - 10 * 10 + 80 = 100 - 100 - 100 + 80 = -20. Since it's -20, so option B is also not correct.

And finally, let's test option D: P = 10, Q = 6. Substituting into the original equation: (10 * 6) - 10 * 10 - 10 * 6 + 80 = 60 - 100 - 60 + 80 = -20. This does not equal zero either.

Conclusion: None of the Options Match

In conclusion, after carefully substituting the values for P and Q from each option into our demand curve equation, none of the options satisfied the equation correctly. This suggests that there might be an error in the given options or in the problem statement itself. The approach we used – substituting and checking – is the correct method, but in this case, it didn't lead us to a valid solution within the provided choices. Sometimes, in problem-solving, it's important to recognize when the given information doesn't quite fit together. So while we didn't find the perfect answer, we did learn how to approach the problem and eliminate incorrect options, and that's still a victory in the world of economics! Keep practicing, and you'll get the hang of these equations in no time! Keep in mind that we may have made a mistake along the way, so be sure to double-check your work!